Solidify: Give different thickness to specific faces
This is very similar to this question, but the method is a tiny bit different.
Instead of setting only the weak part as your vertex group. Set every vertex of your model at 0.5 weight for instance and set the weak spot at 1. Then when you set the vertex group in the solidify modifier, the weak part will get 2 times more thick than the rest of the mesh. The weight is a direct multiplier to the solidify thickness.
Example in action :